Suppose
you've placed a 'certain' amount of money
in a secret savings account in Switzerland
at r-percent interest... To know
roughly when your capital will be doubled,
you can use the trick of Luca
Pacioli, an Italian monk and mathematician
of the Rinascimento: simply divide the
'magic number' 72 by the
interest rate r and you'll obtain
the number of years you should wait for...
Could you explain why and how it works?

The
problem states that any investment at r-percent
interest per annum will be doubled in
approximately [72 / r] years.

In
the middle of a pond lies a water-lily.
The water-lily doubles in size every day.
After exactly 20 days the pond will be
completely covered by the lily. How many
days were necessary to cover half of the
pond?

The Klein
bottle - devised by the German
mathematician Felix Klein (1849-1925)
- was originally called a 'Kleinsche
Fläche' [Klein surface]
in German but mistranslated into English
as 'Kleinsche Flasche' [Klein bottle].
Well, a Klein bottle is a closed non-orientable
surface that has no inside or outside
(i. e. a fly moving around
in this surface can return to it's
starting point mirror-imaged!). It
can be constructed by gluing both
pairs of opposite edges of a rectangle
together giving one pair a half-twist,
but can be physically realized only
in 4 dimensions, since it must pass
through itself without the presence
of a hole. The image opposite is an
'immersion' of the Klein bottle in
a 3-D space (R3), it shows that it
contains one closed, continuous curve
of self-intersection,
where the 'neck' meets the 'bottle'
(it is where the light green part of
the image meets the dark green part).
This topological object is like a Moebius band without
edges! Actually, it is what you get when you glue 2 Moebius bands along their
edges. Of course, as said before, you need an extra space dimension to do this...
So now, you know why a Klein bottle cannot be filled, because there
is no hole at all in it. But how can we visualize a real Klein bottle without
holes? We can often visualize abstract concepts like this if we shift down one
dimension. Let's think about a 2-dimensional equivalent: suppose we live on a
plane, and want to draw a simple closed curve that looks like a figure eight
(8). The trouble is that it intersects itself,
so it's not a simple closed curve. But if we had a good enough imagination to
picture a third dimension, we could lift one part of the curve a little in that
direction, making a 'bridge' so the figure eight didn't intersect! That's the
sort of thing you have to do to make the Klein bottle work (without holes)...
A question, what happens if we make a 'real' hole in a Klein bottle? We transform
this topological surface into another one called Moebius
slip.
We'll end this article by a joking quotation: "in the topologic
Hell the beer is packed in Klein bottles!".